Neutrosophic Sets and Systems, Vol. 9, 2015
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University of New Mexico
Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies Florentin Smarandache1 1
University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: smarand@unm.edu
Abstract. In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, and elementary neutrosophic algebraic structures. Afterwards, their generalizations to refined neutrosophic set, respectively refined neutrosophic numerical and literal components, then refined neutrosophic numbers and refined neutrosophic algebraic structures. The aim of this paper is to construct examples of splitting the literal indeterminacy
(đ??ź) into literal sub-indeterminacies (đ??ź1 , đ??ź2 , ‌ đ??źđ?‘&#x; ) , and to define a multiplication law of these literal sub-indeterminacies in order to be able to build refined đ??ź-neutrosophic algebraic structures. Also, examples of splitting the numerical indeterminacy (đ?‘–) into numerical sub-indeterminacies, and examples of splitting neutrosophic numerical components into neutrosophic numerical sub-components are given.
Keywords: neutrosophic set, elementary neutrosophic algebraic structures, neutrosophic numerical components, neutrosophic literal components, neutrosophic numbers, refined neutrosophic set, refined elementary neutrosophic algebraic structures, refined neutrosophic numerical components, refined neutrosophic literal components, refined neutrosophic numbers, literal indeterminacy, literal sub-indeterminacies, đ??ź-neutrosophic algebraic structures.
1 Introduction
1.2 Neutrosophic Literal Components
Neutrosophic Set was introduced in 1995 by Florentin Smarandache, who coined the words „neutrosophy� and its derivative „neutrosophic�. The first published work on neutrosophics was [1].
In 2003, W.B. Vasantha Kandasamy and Florentin Smarandache [4] introduced the literal indeterminacy “đ??źâ€?, such that đ??ź 2 = đ??ź (whence đ??ź đ?‘› = đ??ź for đ?‘› ≼ 1 , đ?‘› integer). They extended this to neutrosophic numbers of the form: đ?‘Ž + đ?‘?đ??ź, where đ?‘Ž, đ?‘? are real or complex numbers, and
There exist two types of neutrosophic components: numerical and literal. 1.1 Neutrosophic Numerical Components Of course, the neutrosophic numerical components (đ?‘Ą, đ?‘–, đ?‘“) are numbers, intervals, or in general subsets of the unitary standard or nonstandard unit interval. Let đ?’° be a universe of discourse, and đ?‘€ a set included in đ?’°. A generic element đ?‘Ľ from đ?’° belongs to the set đ?‘€ in the following way: đ?‘Ľ(đ?‘Ą, đ?‘–, đ?‘“) ∈ đ?‘€, meaning that đ?‘Ľâ€™s degree of membership/truth with respect to the set đ?‘€ is đ?‘Ą, đ?‘Ľâ€™s degree of indeterminacy with respect to the set đ?‘€ is đ?‘–, and đ?‘Ľâ€™s degree of non-membership/falsehood with respect to the set đ?‘€ is đ?‘“, where đ?‘Ą, đ?‘–, đ?‘“ are independent standard subsets of the interval [0, 1], or non-standard subsets of the non-standard + interval ] − −0, 1 [ , in the case when one needs to make distinctions between absolute and relative truth, indeterminacy, or falsehood. Many papers and books have been published for the cases when đ?‘Ą, đ?‘–, đ?‘“ were single values, or đ?‘Ą, đ?‘–, đ?‘“ were intervals.
(đ?‘Ž1 + đ?‘?1 đ??ź) + (đ?‘Ž2 + đ?‘?2 đ??ź) = (đ?‘Ž1 + đ?‘Ž2 ) + (đ?‘?1 + đ?‘?2 )đ??ź (1) and developed many đ??ź neutrosophic algebraic structures based on sets formed of neutrosophic numbers. The neutrosophic number đ?‘ = đ?‘Ž + đ?‘?đ??ź can be interpreted as: “đ?‘Žâ€? represents the determinate part of numbers đ?‘ , while “đ?‘?đ??źâ€? the indeterminate part of number đ?‘ . 1.3 Notations In order to make distinctions between the numerical and literal neutrosophic components, we start denoting the numerical indeterminacy by lower case letter “đ?‘–â€? (whence consequently similar notations for numerical truth “đ?‘Ąâ€?, and for numerical falsehood “đ?‘“â€?), and literal indeterminacy by upper case letter “đ??źâ€? (whence consequently similar notations for literal truth “đ?‘‡â€?, and for literal falsehood “đ??šâ€?).
Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies
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numerical indeterminacy (đ?‘–) is refined into đ?‘–1 , đ?‘–2 , and đ?‘–3 :
2 Refined Neutrosophic Components In 2013, F. Smarandache [3] introduced the refined neutrosophic components in the following way: the neutrosophic numerical components đ?‘Ą, đ?‘–, đ?‘“ can be refined (split) into respectively the following refined neutrosophic numerical components: ⌊đ?‘Ą1 , đ?‘Ą2 , ‌ đ?‘Ąđ?‘? ; đ?‘–1 , đ?‘–2 , ‌ đ?‘–đ?‘&#x; ; đ?‘“1 , đ?‘“2 , ‌ đ?‘“đ?‘ ; âŒŞ,
(2)
where đ?‘?, đ?‘&#x;, đ?‘ are integers ≼ 1 and max{đ?‘?, đ?‘&#x;, đ?‘ } ≼ 2, meaning that at least one of đ?‘?, đ?‘&#x;, đ?‘ is ≼ 2; and đ?‘Ąđ?‘— represents types of numeral truths, đ?‘–đ?‘˜ represents types of numeral indeterminacies, and đ?‘“đ?‘™ represents types of numeral falsehoods, for đ?‘— = 1, 2, ‌ , đ?‘?; đ?‘˜ = 1, 2, ‌ , đ?‘&#x;; đ?‘™ = 1, 2, ‌ , đ?‘ . đ?‘Ąđ?‘— , đ?‘–đ?‘˜ , đ?‘“đ?‘™ are called numerical subcomponents, or respectively numerical sub-truths, numerical sub-indeterminacies, and numerical sub-falsehoods. Similarly, the neutrosophic literal components đ?‘‡, đ??ź, đ??š can be refined (split) into respectively the following neutrosophic literal components:
3 Refined Neutrosophic Numbers In 2015, F. Smarandache [6] introduced the refined literal indeterminacy (đ??ź) refined as đ??ź1 , đ??ź2 , ‌ , đ??źđ?‘&#x; , with đ?‘&#x; ≼ 2, where đ??źđ?‘˜ , for đ?‘˜ = 1, 2, ‌ , đ?‘&#x;, represents types of literal indeterminacies. A refined neutrosophic number has the general form: đ?‘ đ?‘&#x; = đ?‘Ž + đ?‘?1 đ??ź1 + đ?‘?2 đ??ź2 + â‹Ż + đ?‘?đ?‘&#x; đ??źđ?‘&#x; ,
(4)
where đ?‘Ž, đ?‘?1 , đ?‘?2 , ‌ , đ?‘?đ?‘&#x; are real numbers, and in this case đ?‘ đ?‘&#x; is called a refined neutrosophic real number; if at least one of đ?‘Ž, đ?‘?1 , đ?‘?2 , ‌ , đ?‘?đ?‘&#x; is a complex number (of the form đ?›ź + đ?›˝âˆšâˆ’1 with đ?›˝ ≠0), then đ?‘ đ?‘&#x; is called a refined neutrosophic complex number. Then F. Smarandache [6] defined the refined đ??ź-neutrosophic algebraic structures as algebraic structures based on ⌊đ?‘‡1 , đ?‘‡2 , ‌ đ?‘‡đ?‘? ; đ??ź1 , đ??ź2 , ‌ đ??źđ?‘&#x; ; đ??š1 , đ??š2 , ‌ đ??šđ?‘ ; âŒŞ, (3) sets of refined neutrosophic numbers. Soon after this definition, Dr. Adesina Agboola wrote a where đ?‘?, đ?‘&#x;, đ?‘ are integers ≼ 1 too, and max{đ?‘?, đ?‘&#x;, đ?‘ } ≼ 2 , paper on refined neutrosophic algebraic structures [7]. meaning that at least one of đ?‘?, đ?‘&#x;, đ?‘ is ≼ 2; and similarly đ?‘Ąđ?‘— They were called “đ??ź-neutrosophicâ€? because the refinrepresents types of literal truths, đ?‘–đ?‘˜ represents types of literal ement is done with respect to the literal indeterminacy (đ??ź), indeterminacies, and đ?‘“đ?‘™ represents types of literal falsehoods, in order to distinguish them from the refined (đ?‘Ą, đ?‘–, đ?‘“)-neufor đ?‘— = 1, 2, ‌ , đ?‘?; đ?‘˜ = 1, 2, ‌ , đ?‘&#x;; đ?‘™ = 1, 2, ‌ , đ?‘ . trosophic algebraic structures, where “ (đ?‘Ą, đ?‘–, đ?‘“) -neutrođ?‘Ąđ?‘— , đ?‘–đ?‘˜ , đ?‘“đ?‘™ are called literal subcomponents, or respectively sophicâ€? is referred to as refinement of the neutrosophic nuliteral sub-truths, literal sub-indeterminacies, and literal merical components đ?‘Ą, đ?‘–, đ?‘“. sub-falsehoods. Said Broumi and F. Smarandache published a paper [8] Let consider a simple example. on refined neutrosophic numerical components. Suppose that a country đ??ś is composed of two districts đ??ˇ1 and đ??ˇ2 , and a candidate John Doe competes for the posi4 Neutrosophic Graphs tion of president of country đ??ś. Per whole country, đ?‘ đ??ż (Joe Doe) = (0.6, 0.1, 0.3), meaning that 60% of people voted We now introduce for the first time the general definifor him, 10% of people were indeterminate or neutral – i.e. tion of a neutrosophic graph, which is a (direct or not direct) didn’t vote, or gave a black vote, or a blank vote –, and 30% graph that has some indeterminacy with respect to its edges, of people voted against him, where đ?‘ đ??ż means the neutro- or with respect to its vertexes, or with respect to both (edges sophic logic values. and vertexes simultaneously). We have four main categories But a political analyst does some research to find out of neutrosophic graphs: what happened to each district separately. So, he does a re1) The (đ?‘Ą, đ?‘–, đ?‘“)-Edge Neutrosophic Graph. finement and he gets: In such a graph, the connection between two vertexes đ??´ and đ??ľ, represented by edge đ??´đ??ľ: A-------------------------------B has the neutroosphic value of (đ?‘Ą, đ?‘–, đ?‘“). which means that 40% of people that voted for Joe Doe were 2) đ??ź-Edge Neutrosophic Graph. from district đ??ˇ1 , and 20% of people that voted for Joe Doe This one was introduced in 2003 in the book “Fuzzy were from district đ??ˇ2 ; similarly, 8% from đ??ˇ1 and 2% from đ??ˇ2 were indeterminate (neutral), and 5% from đ??ˇ1 and 25% Cognitive Maps and Neutrosophic Cognitive Mapsâ€?, by Dr. Vasantha Kandasamy and F. Smarandache, that used a diffrom đ??ˇ2 were against Joe Doe. It is possible, in the same example, to refine (split) it in ferent approach for the edge: A-------------------------------B a different way, considering another criterion, namely: what percentage of people did not vote (đ?‘–1 ), what percentage of which can be just đ??ź = literal indeterminacy of the edge, 2 people gave a blank vote – cutting all candidates on the bal- with đ??ź = đ??ź (as in đ??ź -Neutrosophic algebraic structures). lot – (đ?‘–2 ), and what percentage of people gave a blank vote Therefore, simply we say that the connection between ver– not selecting any candidate on the ballot (đ?‘–3 ). Thus, the tex đ??´ and vertex đ??ľ is indeterminate.
Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies
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3) đ??ź-Vertex Neutrosophic Graph. Or a literal indeterminate vertex, meaning we do not know what this vertex represents. 4) (đ?‘Ą, đ?‘–, đ?‘“)-Vertex Neutrosophic Graph. We can also have neutrosophic vertex, for example vertex đ??´ only partially belongs to the graph (đ?‘Ą), indeter-minate appurtenance to the graph (đ?‘–), does not partially belong to the graph (đ?‘“), we can say đ??´(đ?‘Ą, đ?‘–, đ?‘“). And combinations of any two, three, or four of the above four possibilities of neutrosophic graphs. If (đ?‘Ą, đ?‘–, đ?‘“) or the literal đ??ź are refined, we can get corresponding refined neurosophic graphs.
Then, combining the four disjoint parts by none, by one, by two, by three, by four, one gets 2
đ??ś40 + đ??ś41 + đ??ś42 + đ??ś43 + đ??ś44 = (1 + 1)4 = 24 = 16 = 22 . For đ?‘› = 3, one has 23 = 8 disjoint parts,
7 Example of Refined Indeterminacy and Multiplication Law of Sub-Indeterminacies Discussing the development of Refined đ??ź-Neutrosophic Structures with Dr. W.B. Vasantha Kandasamy, Dr. A.A.A. Agboola, Mumtaz Ali, and Said Broumi, a question has arised: if đ??ź is refined into đ??ź1 , đ??ź2 , ‌ , đ??źđ?‘&#x; , with đ?‘&#x; ≼ 2, how to define (or compute) đ??źđ?‘— ∗ đ??źđ?‘˜ , for đ?‘— ≠đ?‘˜? We need to design a Sub-Indeterminacy ∗ Law Table. Of course, this depends on the way one defines the algebraic binary multiplication law ∗ on the set: {đ?‘ đ?‘&#x; = đ?‘Ž + đ?‘?1 đ??ź1 + đ?‘?2 đ??ź2 + â‹Ż + đ?‘?đ?‘&#x; đ??źđ?‘&#x; |đ?‘Ž, đ?‘?1 , đ?‘?2 , ‌ , đ?‘?đ?‘&#x; ∈ đ?‘€},
operators
where đ?‘€ can be â„? (the set of real numbers), or â„‚ (the set of complex numbers). We present the below example. But, first, let’s present several (possible) interconnections between logic, set, and algebra. Logic Disjunction (or) ∨ Conjunction (and) ∧ Negation ÂŹ Implication →
Set Union âˆŞ Intersection ∊ Complement ∠Inclusion ⊆
Equivalence ↔
Identity ≥
Algebra Addition + Multiplication ∙ Subtraction − Subtraction, Addition −, + Equality =
and combining them by none, by one, by two, and so on, by eight, one gets 28 = 256, or 22 = 256. For the case when đ?‘› = 2 = {đ?‘‡, đ??š} one can make up to 16 sub-indeterminacies, such as: đ??ź1 = đ??ś = đ??œđ??¨đ??§đ??đ??Ťđ??šđ???đ??˘đ??œđ??đ??˘đ??¨đ??§ = True and False = đ?‘‡ ∧ đ??š
đ??ź2 = đ?‘Œ = đ??Žđ??§đ??œđ??žđ??Ťđ??đ??šđ??˘đ??§đ??đ??˛ = True or False = đ?‘‡ ∨ đ??š
Table 1: Interconnections between logic, set, and algebra.
đ??ź3 = đ?‘† = đ??Žđ??§đ??Źđ??Žđ??Ťđ??žđ??§đ??žđ??Źđ??Ź = either True or False = đ?‘‡ ∨ đ??š In general, if a Venn Diagram has đ?‘› sets, with đ?‘› ≼ 1, the number of disjoint parts formed is 2đ?‘› . Then, if one combines the 2đ?‘› parts either by none, or by one, or by 2, ‌, or by 2đ?‘› , one gets: đ?‘›
đ?‘›
đ?‘›
đ??ś20đ?‘› + đ??ś2′ đ?‘› + đ??ś22đ?‘› + â‹Ż + đ??ś22đ?‘› = (1 + 1)2 = 22 . (5) Hence, for đ?‘› = 2, the Venn diagram, with literal truth (đ?‘‡), and literal falsehood (đ??š), will make 22 = 4 disjoint parts, where the whole rectangle represents the whole universe of discourse (đ?’°). Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies
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đ??ź4 = đ??ť = đ??§đ??˘đ??Ąđ??˘đ??Ľđ??§đ??žđ??Źđ??Ź = neither True nor False = ÂŹđ?‘‡ ∧ ÂŹđ??š
đ??ź5 = đ?‘‰ = đ??Żđ??šđ?? đ??Žđ??žđ??§đ??žđ??Źđ??Ź = not True or not False = ÂŹđ?‘‡ ∨ ÂŹđ??š
đ??ź6 = đ??¸ = đ??žđ??Śđ??Šđ??đ??˘đ??§đ??žđ??Źđ??Ź = neither True nor not True = ÂŹđ?‘‡ ∧ ÂŹ(ÂŹđ?‘‡) = ÂŹđ?‘‡ ∧ đ?‘‡
Remark One can construct in various ways the diagrams that represent the subindeterminacies and similarly one can define in many ways the ∗ algebraic multiplication law, đ??źđ?‘— ∗ đ??źđ?‘˜ , depending on the problem or application to solve. What we constructed above is just an example, not a general procedure. Let’s see several calculations, so the reader gets familiar: đ??ź1 ∗ đ??ź2 = (shaded area of đ??ź1 ) ∊ (shaded area of đ??ź2 ) = shaded area of đ??ź, or đ??ź1 ∗ đ??ź2 = (đ?‘‡ ∧ đ??š) ∧ (đ?‘‡ ∨ đ??š) = đ?‘‡ ∧ đ??š = đ??ź1 . đ??ź3 ∗ đ??ź4 = (shaded area of đ??ź3 ) ∊ (shaded area of đ??ź4 ) = empty set = đ??ź6 , or đ??ź3 ∗ đ??ź4 = (đ?‘‡ ∨ đ??š) ∧ (ÂŹđ?‘‡ ∧ ÂŹđ??š) = [đ?‘‡ ∧ (ÂŹđ?‘‡ ∧ ÂŹđ??š)] ∨ [đ??š ∧ (ÂŹđ?‘‡ ∧ ÂŹđ??š)] = (đ?‘‡ ∧ ÂŹđ?‘‡ ∧ ÂŹđ??š) ∨ (đ??š ∧ ÂŹđ?‘‡ ∧ ÂŹđ??š) = (impossible) ∨ (impossible) because of đ?‘‡ ∧ ÂŹđ?‘‡ in the first pair of parentheses and because of đ??š ∧ ÂŹđ??š in the second pair of parentheses = (impossible) = đ??ź6 . đ??ź5 ∗ đ??ź5 = (shaded area of đ??ź5 ) ∊ (shaded area of đ??ź5 ) = (shaded area of đ??ź5 ) or đ??ź5 ∗ đ??ź5 = (ÂŹđ?‘‡ ∨ ÂŹđ??š) ∧ (ÂŹđ?‘‡ ∨ ÂŹđ??š) = ÂŹđ?‘‡ ∨ ÂŹđ??š = đ??ź5 . Now we are able to build refined đ??ź-neutrosophic algebraic structures on the set đ?‘†6 = {đ?‘Ž0 + đ?‘Ž1 đ??ź1 + đ?‘Ž2 đ??ź2 + â‹Ż + đ?‘Ž6 đ??ź6 for đ?‘Ž0 , đ?‘Ž1 , đ?‘Ž2 , ‌ đ?‘Ž6 ∈ â„? }, by defining the addition of refined neutrosophic numbers:
Let’s consider the literal indeterminacy (đ??ź) refined into only six literal sub-indeterminacies as above. The binary multiplication law ∗: {đ??ź1 , đ??ź2 , đ??ź3 , đ??ź4 , đ??ź5 , đ??ź6
}2
→ {đ??ź1 , đ??ź2 , đ??ź3 , đ??ź4 , đ??ź5 , đ??ź6 }
(6)
defined as: đ??źđ?‘— ∗ đ??źđ?‘˜ = intersections of their Venn diagram representations; or đ??źđ?‘— ∗ đ??źđ?‘˜ = application of ∧ operator, i.e. đ??źđ?‘— ∧ đ??źđ?‘˜ . We make the following Sub-Indeterminacies Multiplication Law Table:
(đ?‘Ž0 + đ?‘Ž1 đ??ź1 + đ?‘Ž2 đ??ź2 + â‹Ż + đ?‘Ž6 đ??ź6 ) + (đ?‘?0 + đ?‘?1 đ??ź1 + đ?‘?2 đ??ź2 + â‹Ż + đ?‘?6 đ??ź6 ) = (đ?‘Ž0 + đ?‘?0 ) + (đ?‘Ž1 + đ?‘?1 )đ??ź1 + (đ?‘Ž2 + đ?‘?2 )đ??ź2 + â‹Ż + (đ?‘Ž6 + đ?‘?6 )đ??ź6 ∈ đ?‘†6 . and the multiplication of refined neutrosophic numbers: (đ?‘Ž0 + đ?‘Ž1 đ??ź1 + đ?‘Ž2 đ??ź2 + â‹Ż + đ?‘Ž6 đ??ź6 ) ∙ (đ?‘?0 + đ?‘?1 đ??ź1 + đ?‘?2 đ??ź2 + â‹Ż + đ?‘?6 đ??ź6 ) = đ?‘Ž0 đ?‘?0 + (đ?‘Ž0 đ?‘?1 + đ?‘Ž1 đ?‘?0 )đ??ź1 + (đ?‘Ž0 đ?‘?2 + đ?‘Ž2 đ?‘?0 )đ??ź2 + â‹Ż + (đ?‘Ž0 đ?‘?6 + đ?‘Ž6 đ?‘?0 )đ??ź6 + + ∑6đ?‘—,đ?‘˜=1 đ?‘Žđ?‘— đ?‘?đ?‘˜ (đ??źđ?‘— ∗ đ??źđ?‘˜ ) = đ?‘Ž0 đ?‘?0 + ∑6đ?‘˜=1(đ?‘Ž0 đ?‘?đ?‘˜ + đ?‘Žđ?‘˜ đ?‘?0 )đ??źđ?‘˜ + ∑6đ?‘—,đ?‘˜=1 đ?‘Žđ?‘— đ?‘?đ?‘˜ (đ??źđ?‘— ∗ đ??źđ?‘˜ ) ∈ đ?‘†6 . where the coefficients (scalars) đ?‘Žđ?‘š ∙ đ?‘?đ?‘› , for đ?‘š = 0, 1, 2, ‌ 6 and đ?‘› = 0, 1, 2, ‌ 6, are multiplied as any real numbers, while đ??źđ?‘— ∗ đ??źđ?‘˜ are calculated according to the previous SubIndeterminacies Multiplication Law Table. Clearly, both operators (addition and multiplication of refined neutrosophic numbers) are well-defined on the set đ?‘†6 .
Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies
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Florentin Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies